/mg/ - maths general

[Anonymous No.15819034]

Linear programming edition
previously >>15796375
Discuss maths

[Anonymous No.15819247]

>>15819034
>>15811104
>>15818358
Whoever created pic related, this is great.
I would call this the Thomaesian space.

[Anonymous No.15819299]

[Anonymous No.15819301]

>>15819034
What are some fields which integrate well with machine learning besides the obvious ones like non linear optimization and topology?

[Anonymous No.15819304]

how we feeling mathbros

[Anonymous No.15819327]

>>15819304
Dead inside.

[Anonymous No.15819343]

why don't they just make new elementary functions? about time they did

[Anonymous No.15819382]

categorical logic and topos theory? based

[Anonymous No.15819383]

>>15819304
pretty good, for i am currently redpilling myself on intuitionism

[Anonymous No.15819429]

>>15819343
i always thought why they didnt just set the codomain to the image

[Anonymous No.15819477]

>>15819304
Ending a modafinil tolerance break, today's gonna be a good day, I just have to spend it doing homework instead of pursuing schizo math

[Anonymous No.15819506]

Can someone post the /sci/ math reading list?

[Anonymous No.15819549]

>>15819506

[Anonymous No.15819550]

>>15819506
>>15819299

[Anonymous No.15819558]

has anyone tried using ChatGPT to generate exercises? ive had some success doing this with complex analysis, but I think anything more advanced than that and it'll start hallucinating too much for it to be worthwhile (GPT4 btw)

[Anonymous No.15819637]

>>15819549
First book is shit and didn't work.

[Anonymous No.15819646]

>>15819247
Thanks!
It's just some tricks with alpha channel and layers.

[Anonymous No.15819676]

>>15819637
I didn't realize till now that Dale Carnegie has no relation to the steel baron Carnegie. Dale was a salesman-turned-grifter, and changed his last name from Carnegey to Carnegie as part of the grift. No wonder his book didn't work

[Anonymous No.15819690]

>>15819676
The book says shit like "Don't criticize people" but doesn't propose an alternative.
It doesn't teach social thinking, it just teaches you how to fake it till you make it. Good for getting soft skills that might improve your professional career, but bad for if you want to actually make good friends and find happiness.

[Anonymous No.15819746]

>>15819690
In my experience, the only way for two people to become good friends is for them to undergo suffering together. Short of that, they are at best 'drinking buddies.' I assume that in his book 'friend' means more 'person who will do you favors'

[Anonymous No.15819795]

>>15819477
nice dubs, have some schizo math
https://www.youtube.com/watch?v=zmC6XqlBM5U

[Anonymous No.15819851]

Did I do it correctly?

[Anonymous No.15819857]

>>15819558
haven't tried but have considered it
i believe in the future education will be almost entirely AI-driven

[Anonymous No.15819873]

>>15819851
i got the same answer by setting up x-y coordinates and explicitly finding the center with perpendicular bisector

[Anonymous No.15819875]

I want learn number theory(for crypt). Can u say what i should know before jump in?Imagine i only finished school

[Anonymous No.15819883]

>>15819875
number theory is really nice because it has basically no prerequisites. there are some proofs that are easier if you have some basic understanding of group theory (a lot of number theory is essentially the study of cyclic groups in disguise), but all you really need is an understanding of baby-level algebra

[Anonymous No.15819888]

What can I do with math? I've suddenly regained an interest in this intellectual pursuit, but I don't know how I can put to use what I learn.
With compsci at least I could create some shit of my own. Any advice bros??

[Anonymous No.15819893]

>>15819888
it's good for philosophy

[Anonymous No.15819907]

There was a thread on /a/ with a troll math image that was deleted before I could post my answer. I don't have the original problem image.

There are two solutions:

?=25 + 3(sqrt(41)) cm^2
and
?=25 - 3(sqrt(41)) cm^2

Approximately 44.2 cm^2 and 5.8 cm^2 respectively.

It has dimensions:
(19/2 + sqrt(41)/2) cm wide by (11/5 + sqrt(41)/5) cm tall
or
(19/2 - sqrt(41)/2) cm wide by (11/5 - sqrt(41)/5) cm tall

Approximately
(12.7 cm) x (3.5 cm) or (6.3 cm) x (0.9 cm)

You can solve by bounding the bottom left rectangle to an 8 cm^2 area and the upper right rectangle to 32 cm^2.

The triangles are irrelevant.

Once you have done that you can write an equation for where the bottom left rectangle's top right corner has to be given that area. You can do the same for the top right rectangle's bottom left corner.

Then you find where those two meet; the intersection of the two curves.

https://www.wolframalpha.com/input?i=y%3D8%2Fx+and+y%3D6-%2832%2F%2815-x%29%29

[Anonymous No.15819917]

What's the *it* book that everyone uses for Set Theory?
I need a reference for a definition and one I pulled from Kenneth Kunen's is unusual.

[Anonymous No.15819952]

>>15819299
has anyone completed this curriculum

[Anonymous No.15819958]

>>15819917
Naive Set Theory by Halmos
Both Jech and Kunen cut corners and wind up with funky math-flavored poetry. On Kunen page 5, he doesn't define provable. Jech makes a similar error.
On Jech page 157, he writes as if he has defined what it means to tell whether or not "a formal proof of contradiction from Sigma" and as a result the definition he gives for consistent is nonsense.
The errors go all the way back to Gódel.
Look at Godel 1938, "Consistency of the Axiom of Choice" and you'll see the level of rigor that was required for publishing: https://www.pnas.org/doi/epdf/10.1073/pnas.24.12.556

[Anonymous No.15819966]

>>15819958

[Anonymous No.15819968]

>>15819958
and the error in Jech

[Anonymous No.15819976]

>>15819952
All Russian toddlers are proficient in this material by the time they reach pre-school.

[Anonymous No.15819995]

>>15819907
Just go to an archive, nigga.

[Anonymous No.15820022]

>>15819976
out in the tundras maybe
inner city petersburg children learn this before they can walk to the chessboard

[Anonymous No.15820035]

>>15820022
In fact, new mini chalkboards have been developed that expectant mothers can shove into their vaginas along with itty bitty chalk so junior can calculate before birth.
I heard one newborn reciting the quadratic formula before he fully emerged, perhaps he thought the force on his body was related somehow.

[Anonymous No.15820063]

>>15819958
bruh, I just need the definition of well-ordered set

[Anonymous No.15820076]

>>15819888
Shouldn't you ask what kind of math?

[Anonymous No.15820078]

>>15820063
it's in Abstract Algebra by Dummit and Foote '90 edition

[Anonymous No.15820378]

I am so fucking gone guys! How do we make a Turing machine out of logical statements? Has it been done?

[Anonymous No.15820421]

[Anonymous No.15820428]

I have a function [math]f(x, y) = \vert x-y \vert[/math]

Is there any special name for the property where [math]f(x,y) = 0 \iff x = y[/math]?

[Anonymous No.15820475]

>>15820428
similar to the concept of zero divisors in algebra

[Anonymous No.15820532]

>>15820428
https://en.wikipedia.org/wiki/Pseudometric_space

[Anonymous No.15820556]

can a function f(x) go to infinity in finite time without a 1/0 involved? as in any other way than
[eqn]
\lim_{x\rightarrow a}g(x) = 0
[/eqn]
[eqn]
f(x) = \frac{1}{g(x)}
[/eqn]
[eqn]
f(x) = e^{1/g(x)}
[/eqn]

[Anonymous No.15820577]

>>15820532
That's the opposite.
>>15820475
Yeah, I'm just not sure if it's called something. Only thing close I can find is identity of discernibles, but I'm not sure if that describes the function itself or the metric space the points are in

[Anonymous No.15821162]

I fucking love maths, bros. I fucking love proving theorems. I love how it soothes my brain and ends the sizzling of harmful thoughts in my brain. I love how breathing becomes easier, my movements become nimble and my thoughts become light—weightless. I feel as if I become condensed into all of creation and it into me; but like a dandelion, this condensation of abstractions whisks me into the air of ideas when one might have thought me sequestered on a single contemplation.

[Anonymous No.15821168]

>>15820556
f(x) = -ln(x)?

[Anonymous No.15821200]

guys what's the name for the theorem in complex analysis that says a path in an open set can be covered by a finite number of open disks that are subsets of that open set? my textbook calls it "paving lemma" but when i search this up the only results are from the textbook in question

[Anonymous No.15821201]

>>15821200
does this really need a name when it's pretty clear that it holds?

[Anonymous No.15821208]

>>15821201
idk.. sounds better than Theorem 2.7

[Anonymous No.15821255]

>>15820428
If f(x,y) >= 0 equality if and only if x=y, then i have heard the function being referred to as "positive definite" (in the context of metric spaces), but i don't believe this is a universal term

[Anonymous No.15821258]

>>15821255
>using your gay number speak

I'd rather kill myself

[Anonymous No.15821281]

>>15821200
This isnt a theorem specific to complex analysis, this is just the consequence of an open set covering (see compactness)

[Anonymous No.15821379]

>>15819851
didn't look at your solution but: a circle is uniquely determined by three points, solve the coordinates and bingo

[Anonymous No.15821533]

>>15820428
I don't know of any specific name for this property, but it's probably sufficient to say that f "distinguishes distinct points."

If x and y are functions in the structure sheaf of some affine scheme Spec(A), and f is a function which is zero if and only if x and y agree on every point of the scheme, then this property is equivalent to the scheme being reduced, which holds if and only if A is reduced (has no nilpotents). If A is a domain (has no zero divisors per >>15820475), then Spec(A) is both reduced and irreducible which is stronger. So in at least the algebra-geometric sense the relevant algebraic concept is nilpotents, not zero-divisors.

[Anonymous No.15821540]

>>15821379
If you haven't seen it Michael Penn has a video on a nice method for finding the circle intersecting three points:
https://www.youtube.com/watch?v=eeevrPnAB4M

[Anonymous No.15821586]

>>15819304
The future is bright and life is good

[Anonymous No.15821602]

>>15820428
No, it's zero on the diagonal { (x,y) : x = y } and nonzero on the complement of the diagonal. This is how you would introduce the concept as a premise.

[Anonymous No.15821815]

im sorry if this is a double post, but maybe this general is more likely to have some clever people hanging out. can you guys help me solve this puzzle without guesswork and explain it to me how you reached the solution. id appreciate it thanks!

[Anonymous No.15821825]

>>15821815
Mark the 2-3 spaces so that you can see which ones have to be the same. I did that and found a space that gets resolved.

[Anonymous No.15821886]

>>15821825
>Mark the 2-3 spaces
puzzle is nothing but 2-3 spaces, this is why its so tricky theyre are all arranged in a fucked up way. even guessing is pointless. could you elaborate a bit more thanks

[Anonymous No.15822024]

>>15821815
checking cases is the fancy way of saying guesswork

[Anonymous No.15822103]

the 24 year old proover

[Anonymous No.15822155]

Is there a text that helps in seeing analysis in a more unified perspective. I learned algebra, measure, manifolds, fourier, operators, topology, etc. and I seen their uses but is there a subject or text in any of the above that presents things in a combined way. For example, my text on fourier analysis was doing what felt like algebra but they didn't want to use that language. The spaces we studied in functional analysis probably had interesting topology but we focused on classification of hilbert spaces and PDEs.

[Anonymous No.15822160]

>>15819301
Stochastic calculus. Control theory.

[Anonymous No.15822567]

I can't take Calculus anymore, sorry if this post is derivative, but I'm reaching my limit...

[Anonymous No.15822666]

>>15819966
>>15819968
those aren't errors, those authors just didn't include a pointless detour into telling you about hilbert or gentzen proof systems. Any notion of provability that satisfies Godel's completeness will work.

[Anonymous No.15822670]

im prooooving

[Anonymous No.15822801]

how can I prove that cuts are an union of disjoint bonds?

[Anonymous No.15822829]

>>15822567
first year calculus is just gish galloping as much stuff as possible in one semester.
it gets better later

[Anonymous No.15823103]

What do you think Nash equilibrium is here? It's a variant of the prisoner's dilemma from my uni professor and I'm not sure it makes sense.

[Anonymous No.15823107]

>>15823103
It would be where neither of them confess, right? Neither of them would stand to benefit from switching their strategy.

[Anonymous No.15823108]

[math] \text{Jacobson, N.,} \textit{ Basic algebra.} [/math]

[Anonymous No.15823111]

>>15823107
I forgot to mention he says the right answer is Mario confesses and Luigi doesn't confess. The rationale is that Mario can maximise his potential gain by confessing. I disagree and I think it's kind of retarded.

[Anonymous No.15823116]

>>15823111
That is strange, that wouldn’t be a Nash equilibrium because Mario would be better off changing his strategy. That would be the result of each of them trying maximize their own gain irrespective of the other’s choice, but that’s a different question. Maybe he just forgot what a Nash equilibrium is?

[Anonymous No.15823280]

>>15819958
recommending a badly written naive intro to set theory as an alternative to Jech or Kunen because of minor points such as these
>>15819966
>>15819968
makes you look like an absolute moron and I hope no one takes you seriously

[Anonymous No.15823286]

>>15822666
>>15823280
Cope harder. Neither Kunen nor Jech held their work to a standard of mathematical rigor. Their books satisfy no standard of mathematical rigor. Their work is not mathematically rigorous. We do not have a way of finding out what they mean because they use mathematically imprecise language.
You are leading people astray, and you are a FRAUD!

[Anonymous No.15823360]

Can any monotonic curve (in R) composed of elementary functions coincide with a monotonic curve (also in R) composed of non-elementary functions?

[Anonymous No.15823380]

>>15823360
You're asking if a monotonic continuous function f: I - > R "composed of elementary functions" can always be defined in terms of a non-elementary functions? Or are you asking whether the image of f coincides with the image of some non-elementary g: I -> R?

This former is a weird question because usually we talk about a *function* being elementary or not, and clearly f is elementary by definition. The latter is clearly true since f's image is just an interval.

[Anonymous No.15823400]

>>15823360
yes
let I(x) = Integrate[-oo]^x e^(-t^2) dt
note that I is non-elementary and has an inverse, I^-1.
The function sin(x) is elementary.
Let f(x) = I(x) and g(x) = sin(I^-1(x)). We have
sin(x) = g(f(x)
So, the composition of two non-elementary functions, f and g, equals an elementary function, sin.

[Anonymous No.15823408]

>>15823400
oh, oops
I completely forgot about monotonic...the answer is still yes, just replace sin(x) with e^x.

[Anonymous No.15823514]

Say that you want to have five points on the circumference of an ellipse on Cartesian coordinates. Any five points do not define an ellipse this way, but however, any five points selected from the circumference of any specific ellipse do indeed define that ellipse uniquely.

So while thinking about this I came up with this interesting question. What constraints do you need to have for the locations of any five arbitrary points on Cartesian coordinates relative to each other such that it is possible to draw an ellipse through them?

[Anonymous No.15823531]

>>15823514
you would have to play with the equations
the condition is
there exist z,w such that
|a - z| + |a - w| = |b - z| + |b - w| = |c - z| + |c - w| = |d - z| + |d - w| = |e - z| + |e - w|
where a,b,c,d,e are the points given

[Anonymous No.15823587]

does category theory belong more to topology or algebra?

[Anonymous No.15823707]

Hi, could someone hook me up with a decent math problems book for calculus and linear algebra? I understand the theory, but have sometimes hard time to put it into practice at exams :(

[Anonymous No.15823799]

>>15823707
>calculus
Problems and Theorems in Analysis by Pólya and Szegö
>linear algebra
Linear Algebra Problem Book by Halmos

[Anonymous No.15823831]

>>15823799
Thanks m8, I will make you proud.

[Anonymous No.15823908]

>>15821168
No, it has a 1/0 involved by definition
[math]\\ln(x) =\int_1^x \frac{1}{x} [/math]

[Anonymous No.15823969]

>>15820556
Well every function f which blows up in finite time is 1/(1/f) near its pole, so strictly speaking, no; just set g(x) = 1/f(x), and so f = 1/g. If you're willing to accept the taylor series definition tan might fit your bill.

[Anonymous No.15824054]

I think constructivism is neat and the future of mathematics.

[Anonymous No.15824088]

Hi, faggots, currently just about to finish my undergrad. How important is it to have participated/winned maths olympiads to enter a PhD program?
I have a pretty good GPA, but im from a shithole country so i don't know if that is enough.

[Anonymous No.15824095]

>>15824054
I couldn't agree more!

[Anonymous No.15824210]

>>15823286
provable = provable in
https://en.wikipedia.org/wiki/Hilbert_system
= provable in
https://en.wikipedia.org/wiki/Sequent_calculus
again, the authors didn't feel like a pointless detour into proof theory because it's 100% irrelevant to set theory. What they say is mathematically accurate

[Anonymous No.15824253]

>>15824210
you're in denial
you're trying to pretend that errors don't exist
the errors are there
the authors are (implicitly) claiming to have precisely defined their terms, but they haven't
as a matter of mathematical rigor, you are required to state the properties you expect abstract objects to have at the point when they're introduced
what you're trying to do is allow the author to introduce an abstract concept, but then you can drag whatever you want into the definition after the fact
this isn't math; it's a magic trick
you're a stage magician carrying on as if you're a mathematician
authors have to state the properties of the abstract objects they introduce when the abstract object is defined
you don't have the authority to go back and edit the text and ask them to submit the edited manuscript again
you can't just say "well, I don't like what the authors wrote, so I'm going to pretend the authors wrote something else" and then say "well, as far as I'm concerned, the authors wrote something else, so that's the truth" because you're just lying to yourself about what the author wrote

[Anonymous No.15824257]

>>15824253
If the authors had included a statement along the lines of
>for a formal definition of "provability" see [124]
would you be happy?

[Anonymous No.15824298]

>>15824257
you're in denial
you aren't admitting that the authors are cheating when they introduce the concept
if they weren't cheating, they would state the desired properties (and use an adjective that isn't already in use, to avoid confusion) that will be used later on in the text instead of carrying on as if they had introduced what it means for a formula to be provable without actually doing so
do you see the problem??
you haven't stated what abstract properties you are assuming provability to have before you introduce the concept or when you make reference to it; you're trying to do performance magic instead of simply saying what provability is
so, this simply means we don't know what it means for something to be provable because you haven't told us
we still don't know even if you wave your magic wand and say, "Well, because of my magic, now you know."

[Anonymous No.15824300]

>>15824298
I asked a simple yes or no question, what's the answer?

[Anonymous No.15824323]

>>15824300
Then I suppose the answer would have to be yes or no.

[Anonymous No.15824432]

>>15824088
not important
doing well in an olympiad is a very good sign you'll be good at research if you choose to pursue it, but is far from necessary

[Anonymous No.15824447]

>infinitely many axioms
Why didn't people immediately give up on Hilbert systems?

[Anonymous No.15824462]

>>15819851
radius: r
circle points: {0,0} {0,4} {6,6}
center: x,y
rr=xx+(y)(y)
rr=xx+(y-4)(y-4)
rr=(x-6)(x-6)+(y-6)(y-6)
yy = (y-4)(y-4)
yy = yy-8y+16
8y = 16
y = 2
rr=xx+yy
rr=xx+4
rr=(x-6)(x-6)+(y-6)(y-6)
rr=(x-6)(x-6)+16
rr=xx-12x+36+16
rr=xx-12x+52
xx+4=xx-12x+52
4=-12x+52
12x=48
x=4
rr=xx+yy
rr=16+4
rr=20
rr=2*2*5
r=2*sqrt(5)

[Anonymous No.15824759]

>>15819851
You can get there with two equations.
|z+4i|^2 = |z|^2 , |z+6+6i|^2 = |z|^2.
First eq gives z* = z+4i
Second eq gives (1-i)z+(1+i)z* = -12
Plug first into second to get z = -4-2i
|z| = sqrt(20).

[Barkun No.15824762]

2023
>Still taking crackademic math seriously

[Anonymous No.15824910]

i love math so much bros

[Anonymous No.15824993]

>Homotopy type theory
>Turns logical operations, string rewriting programs, symbolic calculus and proofs into surfaces that you can do differential calculus on
If modern mathematics can take anything and transform it into a series of diff.geo exercises, why can't it just skip to the endpoint of this process and create an machine that looks at data (a collection of true statements), creates conjectures and solves the conjectures?

[Anonymous No.15825024]

>>15819034
Hey guys. I've got a question about proofs by contradiction:

I start by making an assumption about an equivalency, then I work the algebra (never dividing by zero!) and I end with the equivalency that 0 = 0. This then shows, that the assumption should hold for ALL values of the variable, if I understand logic correctly.

However, I can then show by inputting actual values, that the equivalency is actually false. I can probably show this arbitrarily many times.

If I can show even a single instance of the equivalency being false, does that not disprove by contradiction the original assumption?

[Anonymous No.15825054]

>>15824054
Unfortunately, it is the past of mathematics.

[Anonymous No.15825111]

>>15825024
>I start by making an assumption about an equivalency, then I work the algebra (never dividing by zero!) and I end with the equivalency that 0 = 0.
This proves exactly nothing.

>However, I can then show by inputting actual values, that the equivalency is actually false.
>If I can show even a single instance of the equivalency being false, does that not disprove by contradiction the original assumption?
As you said, a single counterexample is sufficient to disprove a statement.

[Anonymous No.15825113]

>>15825111
Or to be more precise, if you start by assuming A = B and end with 0 = 0, you have proven
IF A = B, THEN 0 = 0
which is a super obvious and not useful thing to prove.

[Anonymous No.15825116]

>>15819034
I can't believe all the math I learned during undergrad would help me learning math-heavy topics like data structures and algorithms

shit feels good

[Anonymous No.15825150]

>>15825024
To elaborate on what you might have done wrong: Your proof is supposed to END with the statement you're trying to prove, not start with it. Sometimes if you have a "proof" that starts with your statement and ends with 0 = 0 you can reverse the steps and obtain a valid proof. But sometimes you can't.

Examples.

This is not a proof:
6x + 12 = 3(2x + 4)
(6x + 12)/3 = 2x + 4 [divide by 3]
2x + 4 = 2x + 4 [distribute]
2x = 2x [subtract 4]
0 = 0 [subtract 2x]

This is a proof, albeit more complicated than it needs to be:
0 = 0 [reflexive property of =]
2x = 2x [add 2x]
2x + 4 = 2x + 4 [add 4]
(6x + 12)/3 = 2x + 4 [factor]
6x + 12 = 3(2x + 4) [multiply by 3]

This is not a proof:
3x + 1 = 7
0 = 0 [multiply both sides by 0]
And we can't turn it into a proof by reversing it because you can't divide by 0.

Now if we're trying to DISPROVE something, we can start by assuming the statement, and prove a contradiction. What we have proven is

"If [statement], then contradiction"
which is the same thing as "[statement] is false".

Example:
for all real numbers x, 3x + 1 = 7 [assumption]
3(0) + 1 = 7 [specialize to x=3]
3 = 7 [evaluate the left hand side]
contradiction [3 is not 7]
therefore it is not true that for all real numbers x, 3x + 1 = 7 [discharging the assumption]

One thing to take note is that what we have proven is
>it is not true that for all real numbers x, 3x + 1 = 7
which is not the same thing as
>for all real numbers x, it is not true that 3x + 1 = 7
The second statement is false because x could equal 2.

[Anonymous No.15825169]

>>15825150
>for all real numbers x, 3x + 1 = 7 [assumption]
>3(0) + 1 = 7 [specialize to x=3]
>3 = 7 [evaluate the left hand side]
woah

[Anonymous No.15825173]

>>15825169
Yeah I fucked that one up. Let's fix it.

for all real numbers x, 3x + 1 = 7 [assumption]
3(0) + 1 = 7 [specialize to x=0]
1 = 7 [evaluate the left hand side]
contradiction [1 is not 7]
therefore it is not true that for all real numbers x, 3x + 1 = 7 [discharging the assumption]

[Anonymous No.15825176]

>>15825150
>Your proof is supposed to END with the statement you're trying to prove, not start with it.
>>15825150
Wrong.

[Anonymous No.15825181]

>>15825176
Wrong.

[Anonymous No.15825185]

>>15825181
Idiot.

[Anonymous No.15825187]

>>15819304
Pretty good but I'm curious in some things.
I used to participate in math olypiads when I was in high school and I'd like to know if there are more sophisticated tools to tackle those problems after I graduated.
I took the engineering path so I learned calculus, linear algebra, Fourier series and a bit about complex numbers in the way but I don't know where to continue from there.

[Anonymous No.15825190]

>>15825185
Loud guy in a wheelchair who won't stop screaming incoherently.

[Anonymous No.15825192]

>>15825181
>>15825185
>"Gentlemen. You can't correct each other's work in here. This is the math board!"
https://www.youtube.com/watch?v=WI5B7jLWZUc "Dr. Strangelove (2/8) Movie CLIP - No Fighting in the War Room (1964) HD"

[Anonymous No.15825259]

>>15825150
Thank you so much.

[Anonymous No.15825435]

Is there a name for functions that for a given real domain, can output purely real numbers or purely imaginary numbers, but not complex? Or, are there any quirky properties of such functions, compared to functions that map real->real or real->complex?

[Anonymous No.15825460]

>>15825435
The most useful property is if f is holomorphic and real on R then f(z*)=f(z)*

It's seems hard to think of a non-pathological function that is only real on some sub-interval of R.

[Anonymous No.15825500]

>>I'd like to know if there are more sophisticated tools to tackle those problems
First thing that pops to my mind is Burnside's lemma from group theory which makes a lot of counting problems a lot easier.

It depends on your level though. A lot of beginner geometry can be 'bashed' using trig, complex numbers and various other algebraic techniques. But I think there is definitely an emphasis on theory not being that important and problem-solving ability and creativity being the primary thing, especially at the higher level. Thus knowing some obscure theorem shouldn't give you a major advantage (although there are definitely cases where this fails).

[Anonymous No.15825501]

>>15825500 meant for >>15825187

[Anonymous No.15825529]

>>15825460
>if f is holomorphic and real on R then f(z*)=f(z)*
Thanks, wikipedia toime
And isn't f(x) = sqrt(x) only real on a sub-interval of R, if x is real?

[Anonymous No.15825669]

>>15825460
>It's seems hard to think of a non-pathological function that is only real on some sub-interval of R.
square root?

[Anonymous No.15825740]

>[math] i [/math] is imaginary
>[math] \sqrt{i} = \frac{1}{\sqrt{2}} + i\frac{1}{\sqrt{2}} [/math] is complex
What?

[Anonymous No.15825769]

>>15825740
"imaginary" just means "complex with no real component"
if you consider it geometrically, it might make more sense.

multiplication by -1 corresponds to rotation by 180 degrees.
since i is the square root of -1, multiplication by i should correspond to rotation by 90 degrees, so that applying it twice gets you to 180.
similarly, multiplication by the square root of i should correspond to rotation by 45 degrees, so that applying it twice gets you to 90 degrees. but rotating the unit vector by 45 degrees gives you something that's neither wholly real nor wholly imaginary

[Anonymous No.15825786]

>>15825054
more like unfortunately ahead of it's time

[Anonymous No.15825836]

>>15825529
>>15825669
>square root
The branch cut kinda makes it pathological.

[Anonymous No.15825856]

>>15825769
I see the picture you're painting, but that leaves me mystified by the behavior of the complex plane. It's drawn nice and cartesian and vectors act the same under addition, but is a twisted window into hyperbola-land. I've been working on some stuff, I'll have time to pretty it up in latex after I get this homework done

[Anonymous No.15825896]

>>15824762
Omg you guys I just realize that cranks were right all along!

>>15824993
Why indeed.

[Anonymous No.15825967]

If I have some series a-b+c-d+e-f... and so on where each term is a random digit 0-9, what's the probability of a partial sum being 0?
I'm brainlet at combinatorics

[Anonymous No.15826053]

>>15819304
Banging my head against rate distortion theory. Could be worse, could be better.

[Anonymous No.15826064]

Why would a(Q') be equal to a(Q)? I understand we can add any discontinuities to a function, say one at point c in an interval [a,b], then take the area from a to c and from c to b. I also see the intuition behind drawing a circle with a pencil, covering the inside, and then erasing the circle. Same area, but if we have function and we remove every point f(x) without the integral changing (1 layer), why can we not remove n layers? The area would disappear... This is trivial but wtf? Something that has to do with the measure of individual points? The area that is between the lower and the upper integral of a function is snipped without ever considering the graphed line? I don't see it.

[Anonymous No.15826076]

>>15825967
If each element is drawn without conditional dependencies it is the probability of getting n zeros in n draws of the random sequence.

$$
P(x_1 + x_2 + ... + x_n = 0) = P(x_1=0) \cap ... \cap P(x_n=0)
$$

If all of the digits have the same probability of being zero, then it's just

$$
P(x_1 + ... + x_n = 0) = P(x_1=0)^n
$$

If any of the $x_i$ can be negative then your problem is a lot more complicated, but considering you are only allowing 0 to 9 there is exactly one possible sequence by which your partial sum can be equal to zero, namely that all of the individual sequence values are equal to zero.

[Anonymous No.15826100]

>>15826076
>If any of the $x_i$ can be negative then your problem is a lot more complicated
Every other x_i is negative that's why my head hurts

[Anonymous No.15826109]

>>15826100
Oh I'm dumb as fuck and apparently can't read. Also, I'm dumb and can't figure out how to get latex to display properly here.

Okay, basically you need to then need to break it into the events where the sum of the positive numbers are equal to the sum of the negative numbers.

So for example:

p(x_1 - x_2 = 0) = p(x_1 = x_2) = 1- p(x_1 \neq x_2)

If the distribution of the digits is i.i.d and stationary, you'd do the partial sum starting from 2 to be:

\sum_{I=0}^{9}p_i^2

For n=3 you'd need to do the sum of all combinations where x_1 - x_2 + x_3 = 0

There might not be a great way to do this in a general form, but given that sums of random variables produce convolutions, you could probably express it as an expression via a convolution sum expression with even and odd casing.

[Anonymous No.15826118]

>>15826109
Ok chud... Click on the TEX hyperlink on the top left above the [Name] input box.

[Anonymous No.15826122]

>>15826109
Okay. I'll start looking into convolutions. But intuitively, I feel like as the number of digits goes to infinity there should be a 100% probability of a partial sum hitting 0. And from there you could restart and it would be easy to show that the partial sums have to be 0 an infinite amount of times, and then, my end goal is to show that for any number n there will be infinitely many partial sums that equal n. I know that simple random walks have this property but this is different in that there are 10 different "step" lengths and they have to alternate direction. Am I heading in the right direction or am I still an undergrad intuitionlet?

[Anonymous No.15826125]

>>15826109
It's like spoiler tags but you use [math] or [eqn] instead.

[Anonymous No.15826247]

>>15819034
A group can be thought of the structure that automorphisms of a set naturally have. So basically any group is a (subgroup of the) group of automorphisms of some set. And while this isn't used very often in proofs, it often helps understanding things more intuitively when thinking about group elements in terms of actions and symmetry, at least in my experience.
The same is true for rings, since a ring can be thought of the structure that endomorphisms of an abelian group naturally have. But surprisingly, I haven't really seen this mentioned in many textbooks, and it definitely does not help me understand things more intuitively - because rings are usually thought of as "generalised numbers", since the reference point is something like Z/nZ.

Are there any domains or topics where thinking about ring elements as endomorphisms of some abelian group is more natural? I now you can recast many concepts in this view (for example, thinking about modules not as "vector spaces over a ring", but as abelian groups with a ring action into its endomorphism ring), but I am asking about where this actually feels natural and justified, like thinking of groups as codified symmetries.

[Anonymous No.15826324]

>>15825967
It looks like a Markov chain problem to me.
The states are a pair of an integer and some indicator that tells you whether the next term gets added or subtracted. Basically

[eqn]X_0 = (0,+)[/eqn]
[eqn]P(X_{n+1} = (j,+) | P(X_n) = (i,+)) = 0 \\
P(X_{n+1} = (j,+) | P(X_n) = (i,-)) = \begin{cases} \frac{1}{10} & i-j\in\{0,1 \ldots, 9\} \\ 0 & \text{else} \end{cases} \\
P(X_{n+1} = (j,-) | P(X_n) = (i,+)) = \begin{cases} \frac{1}{10} & j-i\in\{0,1 \ldots, 9\} \\ 0 & \text{else} \end{cases} \\
P(X_{n+1} = (j,-) | P(X_n) = (i,-)) = 0
[/eqn]

And your looking for the probability that [math](X_n)[/math] every reach either [math](0,+)[/math] or [math](0,-)[/math] again after the start.

It can be proven that a state is recurrent if the sum of the probabilities to return to it in n-steps from n=1 to infinity diverges. The probability to return to [math](0,+)[/math] in n-steps is zero if n is odd and something bigger than [math]\frac{1}{10n} [/math] if n is even so [math](0,+)[/math] is indeed a recurrent state as the harmonic series diverges.

[Anonymous No.15826334]

>>15825460
x-> [1-i][1+[x/|x|i]x

[Anonymous No.15826356]

in the context of imaginary numbers, is starting to make more sense to me as a rotation matrix.

[Anonymous No.15826425]

wouldn't the completeness of reals imply the existence of a number strictly bigger than 0.999... and strictly smaller than 1?

[Anonymous No.15826487]

yes its 0.999...5

[Anonymous No.15826512]

>>15826425
if you have to ask this question, you don't understand what 0.999... signifies

[Anonymous No.15826528]

>>15820556
yes, the pole of the Riemann zeta function at s=1

[Anonymous No.15826546]

>>15826334
>x/|x|
You might as well just use piecewise defined functions.
I guess you think it is not pathological since you used "basic" operations.

[Anonymous No.15826552]

>>15826546
take the L and move on

[Anonymous No.15826558]

>>15826552
There was no W or L.
These autists became fixated on trying to provide examples to a comment made in passing without first even asking what "non-pathological" could mean or even addressing what the original question was.

[Anonymous No.15826616]

>>15826512
my question is about completeness of reals, not whether 0.99... = 1, i am trying to see whether my understanding is correct.

the completeness of reals means there are no holes in R, there is a real between any two reals or for ever bounded subset of R there is a sup and inf in R for that subset.
let's say we have this subset of R:
[eqn]a_0=0.9\\
a_{n+1}=a_n+\frac{a_n}{10}\\
\{a_n\mid n\in\mathbb{N}\}[/eqn]
would the sup for this set be 1 or something else?
likewise if we had some sequence that converges to one from above but not quite would the inf also be 1?

[Anonymous No.15826619]

>>15826616
>there is a real between any two reals
correction: there are uncountably many reals between any two reals

[Anonymous No.15826620]

>>15826616
>would the sup for this set be 1
yes
>likewise if we had some sequence that converges to one from above but not quite would the inf also be 1
also yes, just like how the infimum of the set defined by 1/n for natural n is 0 even though 0 is not actually an element of the set

[Anonymous No.15826636]

>>15826620
the sup is 1 which is not in the set, what is the max value of the set?
why isn't it 0.999...?

[Anonymous No.15826638]

>>15826636
>what is the max value of the set
There is no maximum value of the set.
>why isn't it 0.999...?
because of the ...
i.e. there are infinitely many 9s. It never actually achieves that value, it only tends to it. If you name any element of the set, I can name a bigger one that's the same thing but with one more 9 slapped onto it

[Anonymous No.15826644]

>>15826425
No, because 0.999... and 1.000... are the same number. Look up any introductory calc course where they define the reals.

[Anonymous No.15826674]

>>15826644
>are the same number.
meaning what?

[Anonymous No.15826684]

>>15826674
0.999... = 1.000...

[Anonymous No.15826689]

>>15826684
>=
meaning what?

[Anonymous No.15826691]

>>15826644
Because the set only contains numbers with a finite string of 9s

[Anonymous No.15826692]

>>15826616
The supremum of the set is infinity because the sequence diverges to infinity.

[Anonymous No.15826696]

>>15826692
>the sequence diverges to infinity.
you know that off right this second

[Anonymous No.15826701]

if number go up it always infinty

[Anonymous No.15826706]

>>15826696
You can easily see that [math]a_n \geq 0.9 + 0.09n[/math] by induction.
It's trivially true for n=0 and assuming it's true for n then for n+1 we have
[eqn]a_{n+1} \geq 0.9 + 0.09n + 0.09 + 0.009n \geq 0.9 + 0.09n + 0.09 = 0.9 + 0.09(n+1)[/eqn]
Since [math]\lim_{n \to \infty} (0.9 + 0.09n) = \infty[/math],
[math]a_n[/math] diverges too.

[Anonymous No.15826717]

>>15826674
Look up any introductory calc course where they define the reals.

[Anonymous No.15826721]

>>15826616
>0.9
>0.9+0.09=0.99
>0.99+0.099=...1.089
a_k = 9*[11^k]/[10^(k+1)]
well shit jigger, check yourself

[Anonymous No.15826725]

>>15826689
I think Bachmann's version is the most eighth grader friendly which is the level this should ideally be taught. Note [math]c_n - a_n \underset{n \to \infty}{\rightarrow} 0[/math] can be replaced with the statement that [math]c_n - a_n[/math] is eventually smaller than any positive rational number.

[Anonymous No.15826727]

>>15826616
You probably wanted
[eqn]a_{n+1} = 0.9 + \frac{a_n}{10}[/eqn]

[Anonymous No.15826769]

What is the physical realisation of a set?
The real, physical objects?
A thought about the real, physical objects? (itself being a physical event in the brain)
Of course, numbers are not physical objects, and sets take numbers as elements.
So, by only referring to physical things, and using concepts which are unambiguously physical, what is the representation of a set in the physical world?

[Anonymous No.15826770]

Does anyone know of a website with interesting problems?
Not the textbook kind, the kind you can show to anyone. If you can't think of a website, something like a book (or similar) works.

[Anonymous No.15826779]

>>15826770
http://www.math.toronto.edu/khesin/papers/CommentsTrivium.pdf
https://projecteuler.net

[Anonymous No.15826844]

https://www.ijcai.org/proceedings/2020/0354.pdf
can you explain how equations 11-14 are derived? 99% fail

[Anonymous No.15827037]

>>15826769
Do you ask the same questions about the playing pieces of board games?

[Anonymous No.15827045]

>>15826770
Try books / websites with programming interview questions. I have no idea why programming interviews use these puzzles. I think it's absurd and illogical. Possibly racist as well, I'm not sure. Maybe just fucking moronic.
https://puzzling.stackexchange.com/questions/230/a-camel-transporting-bananas
https://codingnconcepts.com/puzzle/25-horses-puzzle/

[Anonymous No.15827048]

>>15827037
but i know what the physical realization of a board game, or the rules of a board game, is though

[Anonymous No.15827066]

Taking functional analysis. We defined a compact operator by how it maps bounded sets to paracompact sets. But is this "compactness" equivalent to the topological notion of compactness if we look at the operator norm? I also read that there are other topologies we can have on operators too. Are these equivalent to the operator preserving some property of subsets too?

[Anonymous No.15827187]

>>15826769
Look inside your bottle of retard pills and you will see a physical realization of a finite set

[Anonymous No.15827198]

>>15826769
pencil lead on paper, chalk on blackboard, ink on overhead sheet, illuminated pixel in monitor, sound vibration in air

[Anonymous No.15827202]

>>15827187
I would rather be eaten by a pack of wolves...which also happens to be a physical realization of a finite set
always practice safe sets, boys and girls
don't get eaten by a finite set like I did :(

[Anonymous No.15827219]

>>15826769
There is no such thing.

[Anonymous No.15827499]

Requirements: undergraduate algebra.

[Anonymous No.15827522]

>>15827499
ah shit
I think any permutation tau <- S[n] that commutes with sigma is going to be in Gal(e[sigma]) as well.
Revised conjecture. Let sigma, n be as above. Then Gal(e[sigma]) = { tau <- S[n] : tau sigma = sigma tau }.

[Anonymous No.15827961]

how does one become a quant researcher in a hedge fund, given that one attended the IMO training camp(top 50ish students) in a top10 country

[Anonymous No.15828014]

>>15827961
Apply and do well on the interview.

[Anonymous No.15828030]

Is f(x)=x/x differentiable?

[Anonymous No.15828040]

>>15828030
yes

[Anonymous No.15828059]

>>15828030
no, discontinuity at 0 other wise yes

[Anonymous No.15828065]

Bros... I think I am in love.

[Anonymous No.15828100]

You can't prove in Robinson arithmetic that there are infinite primes. (No induction) but the model of Robinson arithmetic is the natural numbers, yet there must be models with a finite number of prime numbers.

What do such models look like? Can we even know?

[Anonymous No.15828128]

>>15828059
it isn't defined at 0 so the discontinuity doesn't matter

[Anonymous No.15828130]

>>15828100
Any model of robinson arithmetic has infinitely many prime numbers.
You cannot formulate the statement "there are infinitely many prime numbers" in robinson arithmetic.
The closest related statement is "for every n there is an m>n such that m is prime" which means something completely different from there being infinite number of primes.

[Anonymous No.15828167]

>>15828130
>Any model of robinson arithmetic has infinitely many prime numbers.

Anything that is true in all models is provable, no?

[Anonymous No.15828168]

>>15828167
As long as it can be formulated in the theory, yes. That is Godel's completeness theorem.
However, the infinitude of prime numbers cannot be formulated in Q.

[Anonymous No.15828180]

>>15828168
You can formulate for all statements in Robinson Arithmetic. The axioms are all Universally Quantified.

[Anonymous No.15828183]

>>15828180
Go on, formulate the existence of infinite number of prime numbers then. Protip: you can't.

[Anonymous No.15828185]

>>15828183
Oh i get it you're a troll. man thanks for wasting my time.

[Anonymous No.15828203]

>>15828185
You have mental retardation.

[Anonymous No.15828273]

>>15827961
We should publish a list of quants and get them to kill each other for shits and grins.

[Anonymous No.15828275]

>>15828130
I focused on pdes instead of proving stuff, this is interesting.
Why is the statement:
>Set S has no largest prime natural number
not equivalent to:
>there are countably infinitely many prime natural numbers in Set S?
My gut says that if you have one it implies the other so I wanna know what I'm missing.

[Anonymous No.15828306]

>>15828275
for every n there is an m>n such that m is prime is how you say "There are infinite prime numbers" in peano arithmetic and robinson arithmetic.

There is no "is infinite" predicate in first order logic. you can only refer to bijections and shit

[Anonymous No.15828317]

>>15828306
>Definition. Let x be a set. Then x is *finite* if there is no map f : x -> x such that f is injective but not surjective. Otherwise it is infinite.

[Anonymous No.15828323]

>>15828317
>>15828317
And yet you can't say "there are only finitely many elements which satisfy this condition" as a consequence of the compactness theorem

[Anonymous No.15828332]

>>15828323
I know nothing of Robinson arithmetic.You'll have to wait for someone else.

[Anonymous No.15828550]

>>15828306
idk shit about robinson arithmetic can you do indexing to show there is a sequence of primes that is not finite?

[Anonymous No.15828591]

>>15828543
figured I'd also just put this question here
please don't make fun of me, big strong mathematicians

[Anonymous No.15828639]

>>15828591
I tried working it out with M=N=1. I don't know how to calculate a singular value decomposition, though. If you use unknowns for the entries in U and V you can just write out the equations.

[Anonymous No.15828648]

I’m self studying smooth manifolds right now. What would be a good goal to aim for? Is there a certain theorem or problem I should aim to understand? Also, what are some cool things to look at after learning some smooth manifold theory?

[Anonymous No.15828662]

>the set of all computable numbers is countable
this is the final straw

[Anonymous No.15828668]

>>15828648
https://www.youtube.com/watch?v=4TYv2PhG89A "Sade - Smooth Operator - Official - 1984"

[Anonymous No.15828673]

>>15828668
How are jokes like this funny when it could have simply been named something else? If it's just a coincidence, what's funny about it other than there's something else with the same name?

[Anonymous No.15828675]

>>15828662
it's okay, fren
just extend your Turing machine with a symbol for every real in [0,1] and operators for addition, multiplication, subtraction, and division
now every real number is computable

[Anonymous No.15828682]

>>15828673
https://en.wikipedia.org/wiki/N_cell

[Anonymous No.15828725]

>>15828306
>for every n there is an m>n such that m is prime is how you say "There are infinite prime numbers" in peano arithmetic and robinson arithmetic.
The negation of this doesn't imply there are only finitely many primes in the model.

[Anonymous No.15828739]

What are differential equations?
What are partial differential equations and what and how do people model with them?

[Anonymous No.15828749]

>>15828675
that doesn't work thoughbeit.
for a number to be computable you need to be able to calculate it in some concrete way, for example sum of an infinite series.
the set of all such numbers is countable... pretty much that set of all numbers you can think of is countable

[Anonymous No.15828772]

>>15828749
just extend your thought with a parameter that takes a real number
now you have a thought template
>"I'm thinking of the real number x!"
and for every real number x, you have such a thought
that's uncountably many thoughts, in the context where the universe splits into uncountably many parallel universes and then reassembles into a single universe

[Anonymous No.15828775]

>>15828725
There is a model with finitely many primes by the completeness theorem. If there were infinitely many primes in every model, it would be provable from the axioms.

[Anonymous No.15828802]

>>15828772
adding a real number parameter is cheating, that's like saying i can compute any real number then saying real number x is equal to 1*x.
>and for every real number x, you have such a thought
you can't think of a number if you don't know it.
no one in the uncountably many universes can think of these numbers.
if there is any way to concretely visualize, compute, express and/or construct that exact number then the set of all of those is commutable

[Anonymous No.15828820]

>>15828802
I know the value of x.

[Anonymous No.15828827]

>>15828802
moreover, I can think of every real number in the interval [1,oo) in a finite amount of time if time is modeled as a real number and I think of the real number 1/x at time x, for x <- [0,1]

[Anonymous No.15828845]

>>15828739
> differential
The differential is the slope of the ((sometimes) unknown) function with respect to one variable.
> partial
your equations are with the respect to multiple variables
> model
nicely selected multiple variables (aka data) to an appropriate equation to gain useful insight.

[Anonymous No.15828846]

>>15828775
Assuming "for every n there is an m>n such that m is prime" is indeed unprovable from the axioms, there is a model with a largest prime. That doesn't mean the number of primes in that model is finite.

[Anonymous No.15828851]

>>15828749
>for example sum of an infinite series.
*with error bounds on the sum of the remaining terms

[Anonymous No.15828856]

>>15828846
If you define finite as not infinite technically there are a finite number of primes :^)

[Anonymous No.15828863]

Is there a way to get into mathematical proofs as a hobby without taking undergrad maths?
114 IQ midwit here

[Anonymous No.15828867]

>>15828863
Read a textbook.

[Anonymous No.15828869]

Let's say I know the average values in a dataset, but I don't access to the dataset itself.

Example:
Average price of a Blue house: 1000
Average price of a Red house: 1500
Average price of a Green house 900

And let's say I have the average values by location:

Average price in North region: 1200
Average price in South region 1600


Is there a way of knowing that the expected price of a Blue house in the North is? Is there method for this that generalizes across n groups instead of two groups?

[Anonymous No.15828870]

>>15828867
That seems like it would require some degree of explanation. For instance, I have Spivak's Calculus textbook and I find it hard to follow.

[Anonymous No.15828878]

>>15828870
Take a course

[Anonymous No.15828881]

>>15828863
best way to learn proofs is to read proofs and do proofs.
first you need to learn basic first order logic.
then start easy with induction proofs and go to harder stuff relevant to mathematics you are learning. for example epsilon delta.
in my personal opinion proof trees are great for learning what the essence of a proof is even tho they are kinda useless

[Anonymous No.15828918]

>[math]\text{Q.E.D.}[/math]
soul
>[math]\blacksquare /\square[/math]
soulless

[Anonymous No.15829004]

>>15828918
Triforce.

[Anonymous No.15829013]

>>15828918
I prefer ∎ over □ or ◼

[Anonymous No.15829043]

>>15828918
or

[Anonymous No.15829057]

>>15819299
psycho curriculum

[Anonymous No.15829092]

>>15828869
I'm inferring that the 'Average price of a Blue House' is the average across all regions? Off the top of my head, if you can translate this all into a system of linear equations, then the first couple chapters of a Linear Algebra textbook should be able to help you solve these problems for general n. It's intuitive and easy to learn

[Anonymous No.15829160]

>>15828869
Not if that's all the information you have; for example there might not even be any blue houses in the north.
If you knew the proportion of each type of house in each region then you could solve it.

[Anonymous No.15829197]

>>15828869
If I'm understanding your question correctly, then you need 7 more pieces of information that are linearly independent and don't bring in any new unkowns

[Anonymous No.15829229]

>go into maths because I'm a savant and my parents forced me to go to college
>figure that I'll meet more people that are socially awkward like me and I'll have a chance to make friends
>everyone is good at math, my skills are still top level but no one around me is bad at math
>room has a few obvious geniuses but everyone in general probably has a high IQ
>I'm not a novelty at all
>when I was around midwits and low IQs I was a special resource that they wanted, but now I'm pretty fucking worthless
>all the less intelligent people socialize and work together so they don't need me and they already have a genius in their group
>try to demonstrate that I'm useful and really smart by being vocal, usually I never talk and I have a retard accent because I don't talk a lot
>realize I'm actually learning more by talking
>no one cares though
>assigned mentor thinks I'm not serious because I'm a loner and doesn't think I'll get into a good school, and even though I'm a really smart guy he says it's just not enough because everyone in grad school is a really smart guy and the only thing I have going for me is my minor in CS.
>realize the only way is by going to lower years and tutoring them for free "in exchange for a social relationship"
>I'm learning way more and solidifying concepts by tutoring
Is this bad behavior?
I didn't realize so many math students would be so social. No one in my year wants to be a friend, and fuck them. I'm going to gradschool soon anyways so I don't even need friends.
Plus with all this tutoring experience I've been building I think I'm gonna start selling my skills. Also one of my 2nd year friends introduced me to trading. Apparently he wants to be a quant. Apparently you meet a lot of cool people, and being a "superior" or a "senpai" makes it easy for me to have a social relationship with them. It was really easy too. All I did was go to the chair and ask to help him out with the lower years, and he was way more empathetic to me than my mentor.

[Anonymous No.15829249]

Sir this is a Wendy's

[Anonymous No.15829549]

>>15828065
brazilian porn star actress. she doesn't create the vids just reads text from a prompt.
also: b& underage u newfag

[Anonymous No.15829552]

>>15819304
I got kicked out of my masters (on track for PhD afterwards) 7 years ago. I got into a bad way with alcohol and it totally fucked my life up. I think I died back then and I am living in a miserable hell now. I've sobered up but I have nothing worthwhile to do. It's been too long to go back and I can't even afford it. I think about committing suicide daily. Just picking up a textbook and going through it is fun but it's not the same.

[Anonymous No.15829563]

>>15829549
cope

[Anonymous No.15829590]

Every time i feel like i've learned something, i realize i've forgotten it a few months later. How do you guys keep up with your learned maths? For language learning I use flashcards, but that doesn't really work for math I feel like.

What do you guys do? Just occasionally do exercises for problems you haven't faced a lot the last few months?

[Anonymous No.15829594]

Sorry if this is a silly question but I haven't passed highschool math. But why does this equal to 0.575374? Also what do you call these equation with numbers on the top and bottom? Fractions?

[Anonymous No.15829596]

Grad school questions
So I'm in my last year and I planned it all out so I'd have easy classes all year. I'm slated to head off to grad school and I was just wonder what it's like. Honestly I don't really know. I just wanna have an easy life teaching some shit easy classes to retards. My mom wants me to go be an actuary so I can make a lot of money or whatever, but she doesn't get me. I'm a simple guy. I want an easy laid back life. That's what I've been working hard for.

[Anonymous No.15829602]

>>15829594
A rational function
probably to bound it between 0 and 1

[Anonymous No.15829603]

>>15829602
>A rational function
Ah that seems to be it. Thanks!!!

[Anonymous No.15829736]

So why is Jacobson considered a graduate level book? I am finding it much easier than your typical undergraduate recommendations like D&F or Herstein.

[Anonymous No.15829767]

>>15829590
I use flashcards (anki) for math too. Mostly just definitions and theorems, and sometimes a few simple exercises. Also have some cards on intuition for what certain concepts are. Helps me retain the general idea and makes it easier to pick something back up later.

[Anonymous No.15829958]

Would it be easier for more people to understand math if it was presented in the form of programming/code?

[Anonymous No.15829964]

a simple problem from the Regional Mathematics Olympiad(2nd stage for the team selection) held in India yesterday

[Anonymous No.15829969]

>>15829964
[eqn]a_1 = a_2 = \ldots = a_n = 1[/eqn]

[Anonymous No.15829991]

>>15829958
maybe for some.
but there isn't one single way to present math that will serve everyone.
mathematics is a field of multiplicity, e.g., topological problems are seen through algebraic perspectives, combinatorial ones and differential ones.

[Anonymous No.15830006]

>>15829958
Are you the retard who was saying that math symbols should be replaced with code syntax?

[Anonymous No.15830101]

>>15829958
Understanding requires explanation. Usually people are lost on something if they simply don't have the fundamentals, and could muddle through with something like a list and explanation of relevant symbols or examples of how they're used and what they really mean. Since a lot of people I think get stuck on a "symbol-meaning" mismatch where their idea of a thing doesn't match what it's being used for.

"code" does not fix that.

[Anonymous No.15830115]

>>15819304
abandoned, alone, isolated, frustrated, bored, hopeless. I'm a straight A student stuck in classes way below my level, took every testing option still have to grind in classes with students who can't write more than a paragraph that doesn't look gpt generated. My only amusement is writing jokes and using math, or making basic bitch inferences from excel that take a couple seconds.

Society just doesn't seem to give a single shit what you're capable of. I've lost track of how many people I've approached, faculty I've asked, resources I've dug for. I can find nothing to do and nobody who cares and I will be trapped in this hell for at least 4 years if not longer. A hell of buzzwords and trivia that are utterly worthless and meaningless while I keep doing math in between.

Why in the absolute utter hellraising fuck can I not skip it all? Or at least do something in addition? How in the utter fuck is it people look like I've asked about genociding babies just because I'm trying to politely point out everything is WAY below my level and I am desperate for literally any-fucking-thing requiring more than two fucking braincells?

God motherfucking damn changing careers makes me want to suck a 12 gauge

[Anonymous No.15830125]

>>15819304
Bad. I can't work out Euler's proof for the infinity of primes. But I'm guessing the pulsing bloodvessels in more forehead will be relieved if I work out the notation and the overall "gist" of the proof.
Anyone got any ideas?

[Anonymous No.15830142]

>>15830125
What part of the notation is giving you trouble? The set builder notation? Logic notation? Product or sum? It's been a minute but I am pretty sure I can help. I did reply in your other thread but it got shitcanned by a moron pretty much instantaneously after.

[Anonymous No.15830147]

>>15819304
current thinking is electrical impulses transmitted through the body by nerves, maintenance of normal operating parameters for the body, avoiding extremes of hot and cold so as to preserve sensitive bodily tissues, awareness of the environment and sensitivity to activity in the surrounding area

[Anonymous No.15830156]

>>15830125
What book is that?

[Anonymous No.15830169]

>>15830156
I quoted a portion of it and got this searching google books https://www.google.com/books/edition/Proofs_from_THE_BOOK/2iI9BAAAQBAJ?hl=en&gbpv=0

"Proofs from THE BOOK" by martin aigner and gunter ziegler

[Anonymous No.15830181]

>>15830125
you can't work it out because you didn't tell us where you got stuck, Anon...we're happy to help you work it out, but we can't read your mind :(

[Anonymous No.15830225]

>>15830142
>>15830181

Appreciated. I'm trying to methodically decode it bit by bit and I also don't want to waste anyone's time.
My guess is that the overall proof is that a function counting the number of primes that are lesser than or equal to x produces a sequence when the natural numbers are inputted into it that is "unbounded". In this context, meaning that it has no upper limit. That is to say, there is no final highest number of primes this sequence reaches and stops at. Which means that primes are infinite.
To show that the output of this sequence is unbounded, we select a sequence with smaller values that is nonetheless also unbounded.
The sequence that we select for this purpose is the output of the harmonic series, as each new n is introduced, which is unbounded.
We then show that the number of prime numbers (minus one) smaller than or equal to x is greater than the value of this sequence. How we do this, I am not sure.
It seems to be done in multiple steps, showing
-First, that the sum of all 1/ms, where ms are all the natural numbers with prime divisors only smaller than x, is larger than the harmonic series with natural number inputs up to x. I think this makes sense, given that it would be possible to make a 1/m where m = 2 and where m = 2*2 and where m = 2*2*2, forever, in addition to 1/2 + 1/3 + 1/4 etc. Basically, the ms could be larger than n, which summa 1/m could contain all the 1/ns up to and including n=x as well. So summa 1/m should be larger than summa 1/n for any given x.

Then I completely fail to follow the logic or the notation.

[Anonymous No.15830227]

>>15830142
>>15830181

I translate it the next, really confusing part as follows:

m can written as a prime to a power times another prime to a power etc. (so long as the primes being taken to powers are smaller than or equal to x)
This means that summa 1/m = (product of all the prime numbers smaller than x)*(sum of all 1/primes taken to the various powers that prime factors were taken to to make the ms)
Sum of 1/p^k (sum of 1/all the primes that are smaller than x taken to all natural number powers and also the power of 0?) is a geometric series (a series where each additional element is produced by multiplying the previous element) where the number doing the multiplying is 1/p. (1/?)
Hence (unclear why) log x smaller than or equal to product of 1/(1-1/p) where p is each prime number smaller than or equal to x. This can be somehow re-expressed as something to do with p_k?
p_k is somehow clearly greater than or equal to k + 1, thus [some inequalities and equalities that appear to express the same idea]
and therefore
log x - 1 is smaller than or equal to the number of primes smaller than or equal to x. Therefore, infinite primes.

Why we need log x as opposed to just the harmonic series, I don’t know.

[Anonymous No.15830253]

>>15830225
>>15830227
I don't get the proof either, Anon. I'd just ignore it and use the Euclid proof. I can't follow his notation.
Here is the usual proof.
Thm. There are infinitely many primes.
Proof. Assume on the contrary that the finite set P is the set of all primes. Let n = ΠP + 1. We see that n is equivalent to 1 modulo p for all primes p, so no prime divides n. Now suppose m | n. Since m is not prime, it is composite, so it has a prime factor q. Since q | m, we have q | n, a contradiction. This shows that no number in the set {1,2,...,n-1} divides n, so n is prime, a contradiction. QED

[Anonymous No.15830259]

>>15830253
oh, one last thing
we know that n </- P because n > p for all p in P

[Anonymous No.15830262]

>>15830253
I appreciate it man. I'm trying to make a video for YouTube going through every proof in that book. My experience (now on proof 4) has been that they'll all eventually made sense, with enough thought and research.
I'll try another anon's suggestion and go through it here
https://en.wikipedia.org/wiki/Euclid%27s_theorem#Euler's_proof
I've found that reading two versions of the same proof can make it clear.

[Anonymous No.15830284]

>>15830227
>Sum of 1/p^k (sum of 1/all the primes that are smaller than x taken to all natural number powers and also the power of 0?) is a geometric series (a series where each additional element is produced by multiplying the previous element) where the number doing the multiplying is 1/p. (1/?)
>Hence (unclear why) log x smaller than or equal to product of 1/(1-1/p) where p is each prime number smaller than or equal to x.
in general, [math]\sum_{k=0}^{\inf} \frac{1}{x^k} = \frac{x}{x-1}[/math] when |x|>1. This last term is usually stated in the equivalent form [math]\frac{1}{1-\frac{1}{x}[/math], since you get it just by multiplying by [math]\frac{x}{x}[/math]
>This can be somehow re-expressed as something to do with p_k?
In plain English, the version with p on its own is saying "we take this over all primes less than or equal to x", while the version with p_k is saying "we take this product over the first, second, ... prime, up to however many primes there are less than or equal to x". It's the same thing with a different notation
>p_k is somehow clearly greater than or equal to k + 1,
the kth prime will always be at least k+1, since 1 isn't a prime and only naturals are even eligible to consider. 2 is the 1st prime (2=1+1), 3 is the 2nd prime (3=2+1), 5 is the 3rd prime (5>4=3+1), etc.

not going to comment on why you need log x specifically over the harmonic series because, frankly, I'm not sure myself

[Anonymous No.15830293]

>>15830262
I don't like that book because it doesn't have exercises.
I don't like youtube because they don't assign or grade math homework.
I don't like youtube because someone decided to go there with a gun.
https://en.wikipedia.org/wiki/YouTube_headquarters_shooting
Why not just enroll at a college and take a math class from a prof?
Only way to learn is to fork over money for a grader.
Only way to get a grader is to choose a reputable institution of higher education.
Think of it this way: it isn't an IQ test, but you're still paying someone to tell you that you're dumb.
It's essentially masochism / torture / running a marathon.
These "self-learning gentleman mathematicians" should date undergrad qt3.14 and look at her math homework to get an idea of the level of dedication involved. They've all forgotten the pain and trouble of learning.

[Anonymous No.15830302]

>>15819304
Feeling good, finished a paper on modular representations. Shits so cool I love Jordan Canonical form I love hopf algebras I love reduced lie algebras I love group cohomology it's just so fun :)
I remember taking undergrad real analysis thinking that shit was hard but it's good for me to learn well. Turns out it's a dead end for brainlets who think schemes are too abstract. Algebra is the way. Why did I waste so much time learning analysis? That shit belongs in the 19th century lmao

[Anonymous No.15830325]

>>15830293
>I don't like youtube because someone decided to go there with a gun.
what?

[Anonymous No.15830343]

>>15830325
https://youtu.be/_UiL7ErcPf0?t=308

[Anonymous No.15830441]

>>15830125
>>15830225
>>15830227
I don't know how much it'll help but I can offer a proof skeleton:

We're going to prove that the number of primes is infinite by showing that <math>\pi (x)</math> tends to infinity as <math>x</math> tends to infinity (strictly speaking: that for any <math>N</math> we can find a natural number <math>x</math> such that <math>\pi(x) > N</math>; i.e. that the number of primes is greater than any finite number).

STRATEGY: We're going to bound a function we know diverges (log) above by a series of functions, each an upper bound for the next, with the last in the chain being <math>\pi + 1</math>; since log diverges, we'll be able to conclude <math>\pi</math> diverges.

We start with the naïve upper bound for log: the "upper step bound" <math>\log x \leq 1 + 1/2 + \cdots 1/n</math>. [Do you understand this bound?]

INSIGHT 1: Every number 1 through <math>n</math> can be decomposed into prime factors, all of which prime factors are less than <math>x</math>. So <math>\{ 1, 2, \cdots, n\}</math> is a subset of <math>S = \{ n \in \mathbb{N} \, | \, \text{all of n's prime factors are less than}\, x\}</math>. So <math>1 + 1/2 + 1/3 \cdots 1/n</math> is STRICTLY LESS than <math>\sum_{n \in S} \frac{1}{m}</math>.

INSIGHT 2: Consider the (formal) expression <math> (1 + 2 + 4 + 8 + \cdots)(1 + 3 + 9 + \cdots)(1 + 5 + 25 + \cdots)\cdots(1 + q + q^2 + \cdots) </math>, where <math>q</math> is the largest prime less than <math>x</math>. Convince yourself (by expanding) that every element of <math>S</math> is present exactly once in this sum. Therefore we can express the previous sum over S as a product <math>\prod_{p \leq x} \left( \sum_{k\geq 0} \frac{1}{p^k} \right)</math> where all the p's are prime.

1/2

[Anonymous No.15830450]

>>15830441

INSIGHT 3: It's well known [Google geometric series] that if <math>p>1</math> then <math>\sum_{k\geq0}\frac{1}{p^k} = \frac{1}{1-\frac{1}{p}}</math>. We then do some simple manipulations as shown in the book to make this into a friendlier form. [the book writes "the kth prime" as <math>p_k</math>] <math>\prod_{k=1}^{\pi(x)} \frac{p_k}{p_k-1}</math>. This is now our upper bound for log.

INSIGHT 4: The kth prime is greater than k, so we can bound the expression in the product above by an expression involving <math>k</math>. Since we've found an upper bound of the summands, we've found an upper bound for the sum: <math>\prod_{k=1}^{\pi(x)} \frac{k+1}{k}</math>. This expression plainly reduces to <math>\pi(x) + 1</math> [convince yourself by doing it manually for small x!]. Therefore <math>\pi(x) + 1</math> is an upper bound for log and we are done.

Does that help?

2/2

[Anonymous No.15830453]

>>15830441
I don't know how much it'll help but I can offer a proof skeleton:

We're going to prove that the number of primes is infinite by showing that [math]\pi (x)[/math] tends to infinity as [math]x[/math] tends to infinity (strictly speaking: that for any [math]N[/math] we can find a natural number [math]x[/math] such that [math]\pi(x) ] N[/math]; i.e. that the number of primes is greater than any finite number).

STRATEGY: We're going to bound a function we know diverges (log) above by a series of functions, each an upper bound for the next, with the last in the chain being [math]\pi + 1[/math]; since log diverges, we'll be able to conclude [math]\pi[/math] diverges.

We start with the naïve upper bound for log: the "upper step bound" [math]\log x \leq 1 + 1/2 + \cdots 1/n[/math]. [Do you understand this bound?]

INSIGHT 1: Every number 1 through [math]n[/math] can be decomposed into prime factors, all of which prime factors are less than [math]x[/math]. So [math]\{ 1, 2, \cdots, n\}[/math] is a subset of [math]S = \{ n \in \mathbb{N} \, | \, \text{all of n's prime factors are less than}\, x\}[/math]. So [math]1 + 1/2 + 1/3 \cdots 1/n[/math] is STRICTLY LESS than [math]\sum_{n \in S} \frac{1}{m}[/math].

INSIGHT 2: Consider the (formal) expression [math] (1 + 2 + 4 + 8 + \cdots)(1 + 3 + 9 + \cdots)(1 + 5 + 25 + \cdots)\cdots(1 + q + q^2 + \cdots) [/math], where [math]q[/math] is the largest prime less than [math]x[/math]. Convince yourself (by expanding) that every element of [math]S[/math] is present exactly once in this sum. Therefore we can express the previous sum over S as a product [math]\prod_{p \leq x} \left( \sum_{k\geq 0} \frac{1}{p^k} \right)[/math] where all the p's are prime.

[Anonymous No.15830454]

>>15830453
INSIGHT 3: It's well known [Google geometric series] that if [math]p]1[/math] then [math]\sum_{k\geq0}\frac{1}{p^k} = \frac{1}{1-\frac{1}{p}}[/math]. We then do some simple manipulations as shown in the book to make this into a friendlier form. [the book writes "the kth prime" as [math]p_k[/math]] [math]\prod_{k=1}^{\pi(x)} \frac{p_k}{p_k-1}[/math]. This is now our upper bound for log.

INSIGHT 4: The kth prime is greater than k, so we can bound the expression in the product above by an expression involving [math]k[/math]. Since we've found an upper bound of the summands, we've found an upper bound for the sum: [math]\prod_{k=1}^{\pi(x)} \frac{k+1}{k}[/math]. This expression plainly reduces to [math]\pi(x) + 1[/math] [convince yourself by doing it manually for small x!]. Therefore [math]\pi(x) + 1[/math] is an upper bound for log and we are done.

Does that help?

[Anonymous No.15830459]

>>15830441
>>15830450
use square brackets, not angle

[Anonymous No.15830461]

>>15830459
Cheers mate, I am a retard.

[Anonymous No.15830477]

>>15830461
not either anon, but I tried helping him out earlier. Big thanks for continuing to help him. Lots of trolls out today.

[Anonymous No.15830480]

>>15830125
if I were grading this proof, I would take off a point for not proving that
>Sum 1/m
converges, even though it does, given the assumptions (because the partial sums are bounded and monotonically increasing). These details are usually considered too pedantic for textbook readers, however.

[Anonymous No.15830598]

>>15829736
>So why is Jacobson considered a graduate level book?
because it covers graduate level algebra?
>I am finding it much easier than
books aren't supposed to be difficult

[Anonymous No.15830924]

>>15819034
>do dozens of proofs with determinants
>gain zero new understanding of why they work
why are determinants like this?

[Anonymous No.15830933]

>>15830924
because the determinant is the volume of the n dimensional unit interval when it has been transformed by mapping basis vectors to columns in the matrix
intuitively this is why the product of the determinants is the determinant of the product
trace is the same way imho

[Anonymous No.15831049]

>>15830924
read Axler

[Anonymous No.15831597]

>>15830480
the fact you'd raise such a stupid point is why you're not grading or writing a nice book like this

[Anonymous No.15831598]

>>15831049
dogshit book

[Anonymous No.15831624]

>>15830454
Their interpretation of Euler's proof still looks like gobbeldygook to me. However, I found this clip from The Great Courses that actually explains (at least their version of) Euler's proof perfectly:
https://www.youtube.com/watch?v=r5F8fZS8bRU
I need to fill in a couple of details, but I'll see if it ends up resembling the proof in the book.
I think, reading through your telling of the proof, that I can make it all make sense.
I don't really see how pi(x) + 1 = anything in the proof. I don't see how we've produced a formula for counting the number of primes lesser than or equal to x.

[Anonymous No.15831659]

Any ideas for my bachelor thesis in math?
I still can't decide

[Anonymous No.15831684]

>>15830598
Its first 200 pages are basically what is covered by an undergraduate book. And Dummit & Foote also covers graduate level topics in a lot of sections yet is recommended to undergrads.
>books aren't supposed to be difficult
I mean it explains much more than undergraduate books. Just compare the treatment of dihedral groups. One would expect the opposite from a supposedly "graduate level" book.

[Anonymous No.15831687]

>>15830125
how are so many anons in a math general getting filtered by this proof?

[Anonymous No.15831764]

wtf is his problem??

[Anonymous No.15831770]

>>15830454
I think I actually get the entire thing now.
The biggest hurdles were not pre-knowing that that type of series can be converted to that type of fraction.
Also not knowing that there's that relationship between the harmonic series and the negative powers of primes added together and then multiplied together. The notation as it stood wasn't quite enough that I was able to guess what it meant.
Then a couple of minor misconceptions like not knowing what the n was meant to mean in n <_ x <_ n + 1.
Appreciate the help.
I wasn't able to understand solely based on your explanation alone, but it was actually useful to be able to check it and find that I was getting things right where there was ambiguity.
Thank you.

[Anonymous No.15831774]

>>15831687
They don't believe in THE BOOK.

[Anonymous No.15831775]

linear prog

[Anonymous No.15831784]

What would Erdős think of TikTok?

[Anonymous No.15831789]

I'm sure there's some explanation you'll be able to give me that I won't understand, but I saw something a while back that gave me pause.
It was:
-6^2 = -36
Now, I was taught in high school that squaring a negative number results in a positive number, so I saw this as equaling 36. I checked Wolfram Alpha and it told me it was -36, most people I've brought it up to who I feel like should be able to offer an explanation says it's -36 because the -6 isn't in parentheses, and the - sign actually acts as a -1, so the actual equation being presented is -1*6^2, resulting in -36, I see how that conclusion is reached. I contend this is stupid because -1*6^2, using the logic to reach that point is actually -1*1*6^2 and so on, ad infinitum. Why can't -6 just be -6?

[Anonymous No.15831791]

>>15831789
>I contend this is stupid because -1*6^2, using the logic to reach that point is actually -1*1*6^2 and so on, ad infinitum
What's the problem with this?
Nothing changes when you multiply it by 1 by definition. This is true, and it's also -1*1*1*6^2, -1*1*1*6^(2*1), -1*1*1*(6*1)^(2*1), etc.

[Anonymous No.15831795]

>>15831791
While I suppose there's nothing wrong with it, I just find it silly that -1 is allowed to exist as a negative number, but -6 is only allowed to exist as a negative number as a function of -1.

[Anonymous No.15831803]

>>15831795
It's a notational thing more than anything, and nobody would be having that problem if we had a distinct symbol for negatives versus positives (outside of, again, multiplying them by -1). Mostly just a product of the fact that our numerical system is centred around 10, a positive number
If we didn't have a distinct way to represent 6 and instead had to write it as 2*3 or (1+1+1+1+1+1) or something similar, you'd be seeing similar considerations pop up

[Anonymous No.15831805]

>>15831803
Perhaps all results should be expressed with ...*1 at the end.

[Anonymous No.15831814]

>>15831805
Or better yet, bracket all results with *˙˙˙ and ...*1

[Anonymous No.15831816]

>>15831805
the problem with that suggestion is that you end up in a situation where you "should" add another *1 to it, which just leads to an infinite loop
Might as well as suggest that -6 be represented as 6*(-1)*(-1)*(-1), and then as 6*(-1)*(-1)*(-1)*(-1)*(-1), and then...
again, it's just a relic of the fact that our math is centred around a positive base. If we used base -10 instead, where even positions count negatively, then "14" (i.e. -10*1+1*4=-6) would cover the problem entirely and 14^2 would be entirely unambiguous

[Anonymous No.15831819]

>>15831816
Oh well, it's not like it's going to have much of an effect on me as a biochemist.

[Anonymous No.15831825]

>>15819299
>highschool
>introduction to quantum mechanics, topology
nigger what

[Anonymous No.15831833]

>>15831789
I guess the problem is that choosing to have anything as the default meaning also means that you will have to go out of your way to specify other things.
So, for example, we typically take -6^2 to mean -(6^2), which saves us from needing to actually put down those brackets around 6^2, but also means that we need to bracket (-6)^2 when we mean (-6) squared.
If the convention was that -6^2 meant the square of negative six, then we would need to use brackets to indicate when we meant -(6^2).
So no matter what you choose to be the default meaning, you need to use brackets when you deviate off the assumed meaning, and that problem endures no matter what the normal way is.
I guess since math is built out (historically, and, to an extent, theoretically) from the positive natural numbers (positive 1, 2, 3, 4, 5 etc.), those have remained the norm which goes un-notated (usually not written as +1, +2 etc.), and the notation system has been built around regarding negative numbers as weird and requiring additional notation.

[Anonymous No.15831854]

>>15828030
There is no reasonable generally accepted definition of differentiability of an equation.

[Anonymous No.15831861]

Do people ever call e and pi the 'conic/triangular' transcendentals? Are there any other useful trannies?

[Anonymous No.15831888]

>>15828030
for nonzero x it is defined and constant, hence differentiable
for x = 0, f(x) is not defined and hence f is not differentiable at zero

[Anonymous No.15831967]

>>15829552
why were you kicked out?

[Anonymous No.15832023]

Was Russell correct? Are mathematics and logic identical?

[Anonymous No.15832025]

>>15831854
Lol.

[Anonymous No.15832117]

>>15830441
AAHHH A SKELETON

[Anonymous No.15832130]

is feeling like punching someone a normal emotion when doing college math?

[Anonymous No.15832141]

>>15832130
depends who you want to punch

[Anonymous No.15832235]

>>15832141
the first person I see after taking my eyes off a problem.

[Anonymous No.15832444]

>>15832235
you say take eyes off instead of completing
im afraid you have small brain syndrome

[Anonymous No.15832468]

>>15831833
We should have gone with Polish notation.

[Anonymous No.15832475]

>>15831789
conventions vary.
https://en.wikipedia.org/wiki/Order_of_operations#Unary_minus_sign

[Anonymous No.15832489]

>>15832444
ain't news to me doc. don't need a diagnosis

[Anonymous No.15833018]

>>15829594
>a level 25 Sven

That sounds so weird to a scandinavian
That's like playing an RPG and coming across a lvl 25 Michael

[Anonymous No.15833167]

How do I mature (mathematically)? As a self learner and mathlet.

[Anonymous No.15833172]

>>15833167
Being comfortable enough in your foundations that you can comfortably work through, understand, and then explain, papers/theorems/proofs in your area of interest.

Or something along those lines.

[Anonymous No.15833271]

>>15833172
That is more what mathematically mature means. But how do best move towards that, especially in regard to building a solid foundation in Linear Algebra and Analysis/Calculus?

[Anonymous No.15833279]

>>15833271
Oh right sorry I have the dumb. I change my recommendations over the years currently I think starting here as a touch stone helps a lot of people, channel also has other playlists related to your interests,
https://www.youtube.com/watch?v=N-X1EU7tHVo&list=PLBh2i93oe2qtbygdXz4u6Mkh7c_hMLBA8&index=1

in conjunction with khan academy and after that any of the innumerable free textbooks/resources either available via archive.org and/or libretext. If further along, mathematical papers themselves. If you want specific recommendations there've been fucking billions of them given around here and warosu would help you pull them up given the sticky is dead.

[Anonymous No.15833373]

>>15829969
correct. here's a detailed solution:
a
a
....a
1
a
(by using AM
GM)_____(I)








Hence
K
K 1
i
i
(K
1)a
1
Ka 



K
K
i
i
i
K
K 1
i
i
Ka
Ka
a
(K
1)a
1
Ka 





So,
n
i
n
n
K
i
i 1
i
K
i 1
K 1
i
n
a
Ka
a
(by using AM
RMS) ______(II)
n
(K
1)a
1











Since equality holds in (II) all ai’s are equal. While equality must hold in (I) implying ai = 1
Hence a1 = a2 = a3 = …….. an = 1

well, can't seem to copy the latex here

[Anonymous No.15833397]

>>15833167
The most amount of progress I've had so far is working through this collection of proofs: http://cslabcms.nju.edu.cn/problem_solving/images/b/b3/Proofs_from_THE_BOOK_%28Fifth_Edition_2014%29.pdf
But mathematicians all write like complete shit. They can't express a single sentence in plain English and then transition you to technical language. They have a deep, unstoppable psychological need to *begin* in obtuse, technical language that will repeat the same symbols and words in different contexts and expect you to infer or simply guess the meaning.
Which is actually doable if you read two or three different versions of the same proof, while trying to get its general gist, while looking up technical terms, in a repetitive loop.
But then you will find it could have been easily and intuitively explained in about 1 minute at the absolute max, and you grow to despise mathematicians and aim to out-explain math to a wider audience simply to punish mathematicians by being better, rather than out of any sense of altruism.
It's impolitic to say so, but fuck mathematicians. I simply won't let them hoard math.

[Anonymous No.15833431]

>>15826247
No algebraists here? Sad bump